%I #44 Jun 02 2017 22:23:41

%S 0,1,1,2,1,2,1,2,2,2,1,3,1,2,2,3,1,3,1,3,2,2,1,3,2,2,2,3,1,3,1,2,2,2,

%T 2,4,1,2,2,3,1,3,1,3,3,2,1,4,2,3,2,3,1,3,2,3,2,2,1,4,1,2,3,3,2,3,1,3,

%U 2,3,1,4,1,2,3,3,2,3,1,4,3,2,1,4,2,2,2,3,1,4,2,3,2,2,2,3,1,3,3,4,1,3

%N Total number of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.

%C Quetian Superfactorization proceeds by factoring a natural number to its unique prime-exponent factorization (p1^e1 * p2^e2 * ... pj^ej) and then factoring recursively each of the (nonzero) exponents in similar manner, until unity-exponents are finally encountered.

%H Alois P. Heinz, <a href="/A106490/b106490.txt">Table of n, a(n) for n = 1..10000</a>

%H A. Karttunen, <a href="/A091247/a091247.scm.txt">Scheme-program for computing this sequence.</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F Additive with a(p^e) = 1 + a(e).

%F a(1) = 0; for n > 1, a(n) = 1 + a(A067029(n)) + a(A028234(n)). - _Antti Karttunen_, Mar 23 2017

%F Other identities. For all n >= 1:

%F a(A276230(n)) = n.

%F a(n) = A106493(A106444(n)).

%F a(n) = A106491(n) - A064372(n).

%e a(64) = 3, as 64 = 2^6 = 2^(2^1*3^1) and there are three non-1 nodes in that superfactorization. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 5. a(65536) = a(2^(2^(2^(2^1)))) = 4.

%p a:= proc(n) option remember; `if`(n=1, 0,

%p add(1+a(i[2]), i=ifactors(n)[2]))

%p end:

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Nov 07 2014

%t a[n_] := a[n] = If[n == 1, 0, Sum[1 + a[i[[2]]], {i,FactorInteger[n]}]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Nov 11 2015, after _Alois P. Heinz_ *)

%o (Scheme, with memoization-macro definec)

%o (definec (A106490 n) (if (= 1 n) 0 (+ 1 (A106490 (A067029 n)) (A106490 (A028234 n))))) ;; _Antti Karttunen_, Mar 23 2017

%o (PARI)

%o A067029(n) = if(n<2, 0, factor(n)[1,2]);

%o A028234(n) = my(f = factor(n)); if (#f~, f[1, 1] = 1); factorback(f); /* after _Michel Marcus_ */

%o a(n) = if(n<2, 0, 1 + a(A067029(n)) + a(A028234(n)));

%o for(n=1, 150, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 23 2017, after formula by _Antti Karttunen_

%Y Cf. A028234, A064372, A067029, A106444, A106491, A106492, A106493.

%Y Cf. A276230 (gives first k such that a(k) = n, i.e., this sequence is a left inverse of A276230).

%Y After n=1 differs from A038548 for the first time at n=24, where A038548(24)=4, while a(24)=3.

%K nonn

%O 1,4

%A _Antti Karttunen_, May 09 2005 based on _Leroy Quet_'s message ('Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003.